In the past, multivariate DPPs, or the so-called multi-type marked DPPs, have been little explored. In this paper, we propose a class of multivariate DPPs based on a block kernel construction. For the marked DPP, we show that the conditions of existence of DPP can easily be satisfied. The block construction allows us to model the individually marked DPPs as well as controlling the scale of repulsion of points having different marks. Unlike other researchers who model the kernel function of a DPP, we model its spectral representation, which not only guarantees the existence of the multivariate DPP, but makes the simulation-based estimation methods readily available. In our research, we adopted a bivariate complex Fourier basis, which demonstrates nice properties such as constant intensity and approximate isotropy within a short distance between the nearby points. The parameterized block kernels can approximate to commonly-used covariance functions using Fourier expansion. The parameters can be estimated using Maximum Likelihood Estimation, Bayesian approach and Minimum Contrast Estimation.